Boolean Algebra (1.4.3)
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a) Define problems using Boolean logic. See appendix 5e of the specification.
b) Manipulate Boolean expressions. Including the use of Karnaugh maps to simplify Boolean expressions.
c) Use the following rules to derive or simplify statements in Boolean algebra: De Morgan’s Laws, distribution, association, commutation, double negation.
d) Using logic gate diagrams and truth tables. See appendix 5e of the specification.
e) The logic associated with D type flip flops, half and full adders.
Boolean algebra is a sub-area of mathematical algebra where the only values that can be represented are TRUE or FALSE (1 or 0). Knowledge of Boolean algebra is fundamental in programming and the design of modern digital equipment, and it is also used in set theory and statistics. Electronic engineering courses will teach Boolean algebra as standard.
A good approach for teaching Boolean Logic would be to start with the idea of logic gates (Content section d) and truth tables, as these are concepts that can be easily covered and can therefore be used to build students’ confidence with the new material. You may choose to combine practice examples and questions with constructing the logic circuits on a simulator such as ‘Logic gate simulator’ by Steve Kollmansberger. (See 'Logic gate simulator' webpage.)
At this stage, you can then introduce the idea of defining problems with Boolean Logic (Curriculum content section a). You can relate these ideas to real life (e.g. having to take Physics and either Geography or History as subject options) or in terms of machines (e.g. the CTRL, ALT and DEL keys should be pressed all at once). You could then start to look at how different logic circuits can have the same output, and how we might choose to define whole logic circuits using only NAND gates to save money, since NAND gates can be used to emulate any other gate.
You could then look at Curriculum content section b, dealing with manipulating Boolean expressions, firstly teaching simple Boolean simplification laws such as basic Boolean identities and drawing truth tables for these. All of this can be found in 'Logic diagrams' resource.
After this you might cover Curriculum content section c– the laws of commutation, association, distribution and double negation. These are much the same as laws in Mathematics such as A+B is the same as B+A and (A+B)+C is the same as A+(B+C). A good idea would be to then teach the laws of DeMorgan’s Law for algebra simplification. A rundown of these can be found in 'Logic diagrams' resource.
You then might choose to cover Karnaugh maps as something students should understand, but mostly students might prefer to use truth tables as a way to check whether they have manipulated/ simplified their expressions correctly. They may use truth tables as a way of simplification itself, as they may be able to draw conclusions by looking at the output.
Flip-flops, and half and full adders are a bit different from usual logic gate examples and are best taught at the end once students have acquired a full grasp of Boolean logic.
Common misconceptions or difficulties students may have
Boolean algebra is quite a complex topic for those who have not encountered it before and will therefore display quickly the range of abilities in the class.
If students struggle with mathematics they may have trouble getting their heads around distributing Boolean expressions or factoring them out. To help students understand these, you could ask them to produce the circuits in simulators or to complete truth tables to prove each law. Students may find that checking their own simplifications with truth tables before being shown the answer gives them the will to find the answer for themselves, as they will know that they are wrong and can revisit their workings to find out why.
In particular, students sometimes find the second distributive rule A+(B.C) = (A+B).(A+C) quite difficult to master as it does not follow Click here Learner Resource 1 Learner Resource 1 7 the same conventions as normal mathematics, where this would not usually factor out. This is why it is important to cover where the rule comes from. The workings for this can be found in 'Logic diagrams' resource.
Students can have trouble recognising where certain rules can be applied. This can sometimes be due to something as simple as the letters being used, such as XY as opposed to AB. In this case, if students really struggle they might want to replace the letters with ones they are more comfortable with, so long as they change them back afterwards!
Students’ learning and understanding will benefit through having worked-through examples to reference and lots of practice. Parallels could be drawn with puzzles such as Sudoku, in that they require you to have lots of practice to become versed in doing the calculations and complete them with speed.
It would be a good idea to weave in and out of this topic with something that is a bit less taxing. It might, for example, be a good idea to cover a topic like 1.5.2 Ethical, Moral and Cultural Issues at the same time so that this topic has a chance to sink in and students’ interest in it is maintained.
This tactic may also give teachers some time to get exemplar questions completed for homework or as lesson starters/plenaries to help students remember and, if any students have large problems with the topic from a mathematical perspective, to provide them with additional help.
Conceptual links to other areas of the specification – useful ways to approach this topic to set students up for topics later in the course
In concept, specification area 1.4.1 Data Types relates to Boolean algebra as both are to do with how computers deal with logic and numbers. However, for simplicity and teaching, both can be treated as separate entities.You would not want to teach both parts at the same time as there would be too much complexity for students to handle all at once.
'Logic diagrams' and 'Symbols' resources can be used together to introduce how you might go between Boolean algebra equations and a circuit diagram. 'Logic diagrams' resource starts off by giving a few examples and then asks some questions that the learners should have a go at answering.
The 'Symbols' resource can be used for learners to cut out and create their own diagrams so that they can have a go at drawing the circuit into their books, creating a Boolean equation from the diagram and then constructing a truth table. They may then go on to fully prove whether their logic is correct by using a program such as Logic gate simulator (see ‘Thinking conceptually’).
'Karnaugh maps' resource is an introduction to Karnaugh maps with some worked examples. You could use this to introduce the topic.
After studying this worksheet you may choose to set some further problems for students. To view some worked example, see 'Examples 1 link'.
'Examples 2' is another link which is a helpful website with examples of how to (and how not to!) properly group cells.
Unless the student has previously studied computer science, it is unlikely that they will have come across logic gates or Boolean logic before, and we are unlikely to encounter them outside the realm of the subject area.
It is suggested that lots of examples are given and, where possible, examples are related to real life as opposed to just 0s and 1s, such as “To launch a missile, the operator must turn the key (logic 1) AND press the button (logic 1)”.
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